Used Cars – Applied ML Project

About the Dataset (Summary)

The Used Car Price Prediction dataset contains 4,009 vehicle listings collected from the automotive marketplace cars.com. Each row represents a unique car and includes nine key attributes relevant to pricing and vehicle characteristics. Dataset is taken from Kaggle: https://www.kaggle.com/datasets/taeefnajib/used-car-price-prediction-dataset

The dataset provides information on:

Brand and model – manufacturer and specific vehicle model

Model year – age of the car, influencing depreciation

Mileage – an indicator of usage and wear

Fuel type – e.g., gasoline, diesel, electric, hybrid

Engine type – performance and efficiency characteristics

Transmission – automatic or manual

Exterior/interior colors – aesthetic properties

Accident history – whether the car has previously been damaged

Clean title – legal/ownership status

Price – listed price of the vehicle

Overall, the dataset offers a structured overview of key features that influence used car valuation. It is well-suited for analytical tasks such as understanding pricing drivers, exploring consumer preferences, and building predictive models for vehicle prices. # Raw data

We load the original CSV directly from the project data folder using here() so paths work regardless of the working directory.

raw_path <- here("data", "raw", "used_cars.csv")
cars_raw <- readr::read_csv(raw_path, show_col_types = FALSE)

Basic structure and summary statistics of the raw dataset:

glimpse(cars_raw)

## Rows: 4,009
## Columns: 12
## $ brand        <chr> "Ford", "Hyundai", "Lexus", "INFINITI", "Audi", "Acura", …
## $ model        <chr> "Utility Police Interceptor Base", "Palisade SEL", "RX 35…
## $ model_year   <dbl> 2013, 2021, 2022, 2015, 2021, 2016, 2017, 2001, 2021, 202…
## $ milage       <chr> "51,000 mi.", "34,742 mi.", "22,372 mi.", "88,900 mi.", "…
## $ fuel_type    <chr> "E85 Flex Fuel", "Gasoline", "Gasoline", "Hybrid", "Gasol…
## $ engine       <chr> "300.0HP 3.7L V6 Cylinder Engine Flex Fuel Capability", "…
## $ transmission <chr> "6-Speed A/T", "8-Speed Automatic", "Automatic", "7-Speed…
## $ ext_col      <chr> "Black", "Moonlight Cloud", "Blue", "Black", "Glacier Whi…
## $ int_col      <chr> "Black", "Gray", "Black", "Black", "Black", "Ebony.", "Bl…
## $ accident     <chr> "At least 1 accident or damage reported", "At least 1 acc…
## $ clean_title  <chr> "Yes", "Yes", NA, "Yes", NA, NA, "Yes", "Yes", "Yes", "Ye…
## $ price        <chr> "$10,300", "$38,005", "$54,598", "$15,500", "$34,999", "$…

Exploratory Data Analysis

We base the EDA on the engineered dataset (data/processed/used_cars_features.csv) that keeps cleaned numeric fields and derived features like age, mileage in thousands, and accident flags.

features_path <- here("data", "processed", "used_cars_features.csv")
cars <- readr::read_delim(features_path, delim = ";", show_col_types = FALSE)

Key descriptive values

Key numeric feature summaries

variable	median	mean	p25	p75	sd	min	max
price_dollar	28000.00	36865.68	15500.00	46999.00	36531.16	2000.0	649999.00
log_price	10.24	10.19	9.65	10.76	0.82	7.6	13.38
age	9.00	10.32	6.00	14.00	5.87	1.0	29.00
milage_k	63.00	72.14	30.00	103.00	53.60	0.0	405.00
horsepower	310.00	331.51	248.00	400.00	120.32	76.0	1020.00

Accident history distribution

accident	n	share
At least 1 accident or damage reported	871	0.28
None reported	2194	0.72

Median listing sits around $28k, with the middle 50% between roughly $15.5k and $47k, while the maximum reaches $650k—explaining the heavy right tail. Median age is 9 years (IQR: 6–14), typical mileage is about 63k miles (IQR: 30k–103k), and horsepower clusters around 310 HP (IQR: 248–400). About 28% of cars report an accident or damage, a meaningful factor for pricing.

Price distribution (raw and log)

Raw prices are extremely right-skewed, with most listings below $80k but a long tail of luxury and exotic vehicles. Modeling on this scale would be dominated by a few high-price outliers.

Log transformation produces a more bell-shaped distribution and stabilizes variance, making linear-style models and visual comparisons more reliable.

Depreciation by age and fuel type

Prices decline with age across fuels. Electric listings start high but show the sharpest early drop; diesel holds comparatively high prices across ages (though the diesel sample is small), and gasoline sits lower overall.

Price spread across top brands

Among the 12 most common brands, Porsche leads on median price, followed by Land Rover and Mercedes-Benz; Volume brands (Toyota, Nissan, Jeep) cluster lower with tighter spreads, while some (Chevrolet, Ford) span broader lineups.

Mileage impact by transmission

Higher mileage correlates with lower prices. We use a loess smoother (not a straight trendline) and cap the x-axis at 250k miles to reduce the influence of extreme outliers; automatics show a steady decline, and the smaller manual subset is noisier but similar in direction.

Horsepower premium

Price rises with horsepower, especially from ~250 HP upward; we cap horsepower at 700 to avoid a handful of ultra-high-HP outliers from distorting the loess smoother, so the trend reflects the bulk of the market rather than extreme sports models.

Accident history effect

Cars with reported accidents trade at a clear discount relative to clean histories, even after log-scaling prices, confirming accident history as an important predictor.

4. Linear regression model

In Section 3 we saw that price is strongly right-skewed and that log(price) has an approximately linear relationship with age. Based on this, we now fit a linear regression model with log(price) as response.

The goal is not primarily to build the most accurate predictor, but to understand which factors drive used-car prices and in which direction.

We model the natural logarithm of the price instead of the raw price because:

• prices are strictly positive and strongly right–skewed,
• taking the log makes the distribution more symmetric,
• the linear model assumptions (linear relationship, constant variance of the residuals) are usually better satisfied on the log scale,
• coefficients can be interpreted approximately as percentage effects.

4.1. Model specification

We use the feature dataset created in the previous step (data/processed/used_cars_features.csv) and fit the following multiple linear regression model on the log-price:

log_price ~ age + milage_k + accident_bin + brand + fuel_type + transmission + ext_col + int_col

Here

• log_price is the natural logarithm of the car price in dollars (the target variable / response).
• age is the car age in years.
• milage_k is the mileage in thousands of miles.
• accident_bin is a binary variable
  (0 = no accident reported, 1 = at least one accident or damage reported).
• brand, fuel_type, transmission, ext_col and int_col are categorical predictors and are represented in the model by dummy variables with one reference category each.
• The error term (residual) captures the remaining variation not explained by the predictors.

The coefficients of this model measure how much the expected log-price changes when we increase a numeric predictor by one unit (or switch a dummy variable from 0 to 1), while keeping all other variables fixed.

4.2 Estimation and overall performance

We implement and fit the model by sourcing the script src/04_model_linear.R, which

1. loads the feature data (used_cars_features.csv),
2. selects the variables listed above and removes rows with missing values,
3. fits the linear model with stats::lm(...), and
4. computes the root mean squared error (RMSE) and \(R^2\) on the log-price scale and adds predictions back to the data.

# Fit the baseline linear regression model and prepare lm_linear_data
source(here::here("src", "04_model_linear.R"))

# Show a compact summary of the model 
broom::tidy(lm_linear) |> 
  dplyr::slice(1:10)  # show only first 10 coefficients

## # A tibble: 10 × 5
##    term          estimate std.error statistic   p.value
##    <chr>            <dbl>     <dbl>     <dbl>     <dbl>
##  1 (Intercept)   11.4      0.115       98.9   0        
##  2 age           -0.0578   0.00188    -30.8   5.45e-181
##  3 milage_k      -0.00599  0.000198   -30.3   3.05e-176
##  4 accident_bin  -0.0801   0.0174      -4.61  4.28e-  6
##  5 brandAlfa     -0.0619   0.146       -0.424 6.71e-  1
##  6 brandAudi      0.225    0.0728       3.10  1.98e-  3
##  7 brandBentley   1.27     0.111       11.4   1.32e- 29
##  8 brandBMW       0.281    0.0688       4.08  4.61e-  5
##  9 brandBuick    -0.172    0.119       -1.44  1.49e-  1
## 10 brandCadillac  0.276    0.0788       3.50  4.77e-  4

4.3 Interpretation of selected coefficients

Because the model is fitted on the log(price) scale, each coefficient can be read approximately as a percentage change in price when we increase that variable by one unit (or switch a dummy variable from 0 to 1), while keeping all other variables fixed. Roughly, a coefficient of −0.06 means “about −6 %”, a coefficient of +0.20 means “about +22 %”, and so on.

Below we interpret a few selected coefficients from the model.

• Age (coefficient ≈ −0.058)
  Holding all other variables constant, increasing the age of a car by one year reduces the expected price by about 6 % (exp(−0.058) ≈ 0.94).
  → Older cars are substantially cheaper, as expected.

• Mileage in thousands (coefficient ≈ −0.006)
  An additional 1,000 miles reduces the expected price by roughly 0.6 %.
  → The effect of mileage is noticeable but smaller than the effect of age.

• Accident history (accident_bin, coefficient ≈ −0.080)
  accident_bin = 1 indicates that at least one accident or damage has been reported. Compared to a car with no accident history (accident_bin = 0), the expected price is lower by about 8 %.
  → Cars with an accident history sell for clearly lower prices.

• Brand examples
  The brand coefficients show how each brand differs from the (omitted) reference brand, holding age, mileage, accident history and all other variables constant.

  • brandBMW (coefficient ≈ 0.28): price is about 30 % higher than for the reference brand.
    → BMW cars are noticeably more expensive, even after controlling for other factors.

  • brandFerrari (coefficient ≈ 1.82): price is more than six times higher (over +500 %) than for the reference brand.
    → Ferrari appears as an extreme premium brand in this dataset.

• Fuel type example (fuel_typeElectric, coefficient ≈ −0.79)
  For purely electric cars, the coefficient corresponds to a price that is roughly 55 % lower than for the reference fuel type, given the same age, mileage, brand etc.
  → In this sample, electric cars are priced clearly below comparable vehicles with the reference fuel type (possibly due to different model mix, range concerns, or incentives on combustion cars).

• Transmission (transmissionManual, coefficient ≈ 0.20)
  Manual transmission has a coefficient of about 0.20, which translates to prices that are roughly 20–22 % higher than for the reference transmission type.
  → Cars with manual transmission are on average more expensive in this sample.

Overall, the signs and magnitudes of the coefficients are plausible: older and higher-mileage cars and cars with accidents are cheaper, while premium brands and certain configurations (e.g. BMW, Ferrari) command much higher prices. With an \(R^2\) of about 0.75 on the log-price scale, the model explains a large share of the variation in used car prices, even though there is still substantial unexplained variability.

4.4. Relationship between age and log-price

To visualise the effect of age in the linear model, we plot log-price against age and add a fitted linear trend line.

The plot shows the expected negative relationship: older cars tend to have a lower log-price. In our model, the age coefficient is about −0.058, which means that, holding all other variables constant, one additional year of age reduces the expected price by roughly 6 % (because exp(−0.058) ≈ 0.94).

4.5. Residual diagnostics

To check whether the linear model assumptions are roughly satisfied, we plot the residuals against the fitted (predicted) log-prices. Ideally, the points are scattered randomly around zero without a clear pattern.

In our plot, the residuals are roughly centred around zero with no strong curvature. There is some increase in spread for higher predicted prices, but overall the linear model assumptions appear acceptable.

4.6 Conclusions (linear regression)

Overall, the linear regression on log(price) provides a simple and interpretable summary of the main price drivers in this dataset. Age and mileage have the expected negative impact on prices, while an accident history leads to a price discount of roughly 8 %. Premium brands such as BMW and especially Ferrari command large price premia, even after controlling for age, mileage and fuel type. The model explains about 75 % of the variance in log-prices, which is quite high for real-world data, but the residual plot also shows that there is still substantial unexplained variability. For a client, this model could already be used as a rough pricing guideline, but more advanced models (e.g. with interactions or nonlinear effects) might capture the remaining structure in the data even better.

5. Generalized Linear Model — Binomial (Logistic Regression)

A binomial Generalized Linear Model (GLM) is appropriate when the response variable is binary.
Here, the response variable is:

• accident_bin
  • 1 = car has had at least one reported accident
  • 0 = no reported accident

The model estimates the probability that a used car has an accident history based on vehicle characteristics.

5.1 Model specification

We model accident history using a logistic regression with a logit link:

\[ \log\left(\frac{P(\text{accident}=1)}{1 - P(\text{accident}=1)}\right) = \beta_0 + \beta_1 X_1 + \dots + \beta_p X_p \]

Model formula:

accident_bin ~ brand + age + milage_k + fuel_type + transmission + price_dollar + horsepower + cylinders

• Response: accident_bin (binary)
• Predictors:
  • Continuous: age, mileage (1,000 miles), price, horsepower, cylinders
  • Categorical: brand, fuel type, transmission
• Link function: logit

5.2 Model fitting and summary

# Source the script that fits the binomial GLM
source(here::here("src", "05_model_glm_binomial.R"))

# Show a full summary of the model 
summary(glm.car)

## 
## Call:
## glm(formula = accident_bin ~ brand + age + milage_k + fuel_type + 
##     transmission + price_dollar + horsepower + cylinders, family = "binomial", 
##     data = data)
## 
## Coefficients:
##                           Estimate Std. Error z value Pr(>|z|)    
## (Intercept)             -7.565e-01  4.880e-01  -1.550   0.1211    
## brandAlfa               -1.102e+00  1.113e+00  -0.990   0.3221    
## brandAudi                2.338e-01  3.826e-01   0.611   0.5412    
## brandBentley             7.038e-02  7.604e-01   0.093   0.9263    
## brandBMW                -1.410e-01  3.655e-01  -0.386   0.6997    
## brandBuick               3.798e-01  6.019e-01   0.631   0.5280    
## brandCadillac            4.799e-02  4.195e-01   0.114   0.9089    
## brandChevrolet          -2.149e-01  3.804e-01  -0.565   0.5721    
## brandChrysler           -6.488e-02  5.726e-01  -0.113   0.9098    
## brandDodge               8.623e-02  4.292e-01   0.201   0.8408    
## brandFerrari             1.980e+00  1.275e+00   1.553   0.1205    
## brandFord               -8.352e-02  3.679e-01  -0.227   0.8204    
## brandGenesis            -1.373e+00  1.095e+00  -1.254   0.2097    
## brandGMC                 1.849e-01  4.318e-01   0.428   0.6685    
## brandHonda              -2.186e-01  4.732e-01  -0.462   0.6441    
## brandHummer              3.534e-01  6.157e-01   0.574   0.5660    
## brandHyundai             1.307e-01  4.424e-01   0.295   0.7677    
## brandINFINITI           -2.661e-01  4.589e-01  -0.580   0.5621    
## brandJaguar             -2.984e-02  5.020e-01  -0.059   0.9526    
## brandJeep               -3.128e-01  4.153e-01  -0.753   0.4513    
## brandKia                -1.944e-01  4.982e-01  -0.390   0.6964    
## brandLamborghini        -1.127e+01  3.235e+02  -0.035   0.9722    
## brandLand               -7.250e-01  4.439e-01  -1.633   0.1025    
## brandLexus               3.271e-01  3.909e-01   0.837   0.4027    
## brandLincoln            -1.301e-01  4.753e-01  -0.274   0.7843    
## brandMaserati           -1.064e+00  7.121e-01  -1.494   0.1352    
## brandMazda              -4.210e-02  5.076e-01  -0.083   0.9339    
## brandMercedes-Benz       8.343e-02  3.713e-01   0.225   0.8222    
## brandMINI               -1.140e+00  6.409e-01  -1.778   0.0754 .  
## brandMitsubishi          1.051e-01  5.975e-01   0.176   0.8604    
## brandNissan              1.616e-01  4.075e-01   0.396   0.6918    
## brandPontiac            -3.582e-01  6.867e-01  -0.522   0.6019    
## brandPorsche             2.548e-01  4.055e-01   0.629   0.5297    
## brandRAM                 3.967e-01  4.589e-01   0.864   0.3873    
## brandRivian             -1.300e+01  3.749e+02  -0.035   0.9723    
## brandSubaru              2.139e-01  4.458e-01   0.480   0.6313    
## brandTesla              -1.094e+00  8.824e-01  -1.240   0.2150    
## brandToyota             -1.868e-01  3.866e-01  -0.483   0.6289    
## brandVolkswagen         -3.929e-01  4.792e-01  -0.820   0.4123    
## brandVolvo              -5.475e-01  5.440e-01  -1.006   0.3142    
## age                     -1.164e-02  1.190e-02  -0.978   0.3281    
## milage_k                 6.995e-03  1.150e-03   6.083 1.18e-09 ***
## fuel_typeE85 Flex Fuel  -1.023e-01  3.112e-01  -0.329   0.7424    
## fuel_typeElectric       -1.286e+00  6.523e-01  -1.972   0.0486 *  
## fuel_typeGasoline       -1.889e-01  2.541e-01  -0.743   0.4574    
## fuel_typeHybrid         -2.298e-01  3.338e-01  -0.689   0.4911    
## fuel_typePlug-In Hybrid  4.227e-01  4.819e-01   0.877   0.3803    
## transmissionManual      -2.091e-01  1.606e-01  -1.302   0.1929    
## price_dollar            -2.169e-05  3.845e-06  -5.642 1.68e-08 ***
## horsepower               1.829e-03  8.796e-04   2.079   0.0376 *  
## cylinders               -3.470e-02  5.214e-02  -0.665   0.5058    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 3658.7  on 3064  degrees of freedom
## Residual deviance: 3306.0  on 3014  degrees of freedom
## AIC: 3408
## 
## Number of Fisher Scoring iterations: 14

5.3 Key Odds Ratios and Interpretation

We scale and exponentiate the coefficients of the binomial GLM to obtain odds ratios, which show how the odds of a car having an accident history change with each predictor.

To improve interpretability, continuous predictors are rescaled before computing odds ratios. This rescaling does not affect statistical significance or model fit, only the unit of interpretation.

The data were scaled as follows:

• milage_k was multiplied by 10.
• price_dollar was multiplied by 1,000.
• horsepower was multiplied by 50.

GLM Binomial: Key Odds Ratios and Percent Change (scaled)

term	odds_ratio	pct_change	p.value	signif
milage_k	1.072	7.245	0.000
fuel_typeElectric	0.276	-72.374	0.049
price_dollar	0.979	-2.146	0.000
horsepower	1.096	9.575	0.038

Interpretation of key predictors:

• Mileage (milage_k): Odds ratio > 1 → Higher mileage slightly increases the odds of a reported accident. Specifically, an additional 10,000 miles increases the odds by ~7%, reflecting increased usage and exposure.

• Price (price_dollar): Odds ratio < 1 → Higher-priced vehicles are less likely to have an accident. Each additional $1,000 reduces the odds by ~2%, suggesting that more expensive cars are generally newer or better maintained.

• Horsepower (horsepower): Odds ratio > 1 → More powerful cars slightly increase accident odds. For example, a 50 HP increase raises the odds by ~9.5%, possibly due to more spirited driving.

• Fuel type – Electric (fuel_typeElectric): Odds ratio << 1 → Electric cars have substantially lower odds (≈ 72 % lower) of reported accidents compared to the reference fuel type. This effect should be interpreted cautiously, as electric vehicles in this dataset are generally newer and may benefit from modern safety technologies.

These odds ratios allow for an intuitive interpretation of how each variable affects accident probability, holding all other variables constant. Most other predictors (brand, transmission, cylinders, age) were not statistically significant in this model.

5.4 Conclusions (Binomial GLM)

The binomial GLM indicates that accident probability is primarily driven by vehicle usage and value. Higher mileage and higher horsepower are associated with increased accident odds, while higher-priced cars are less likely to have a reported accident history. This is consistent with the intuition that cheaper cars tend to be older, more heavily used, and exposed to greater wear and risk. Electric vehicles exhibit substantially lower accident odds in the model. However, this result is likely influenced by sample composition: electric cars in the dataset are generally newer and therefore have had less exposure time to accidents. The observed fuel-type effect may therefore partially reflect vehicle age rather than an intrinsic safety advantage. Overall, the GLM provides a transparent and interpretable baseline model for accident risk. Its linear structure, however, limits its ability to capture nonlinear exposure effects, motivating the use of a Generalised Additive Model in the next section.

6. Generalized Linear Model — Poisson

A Poisson GLM is typically used when the response variable is a count (non-negative integers). Because this dataset does not contain a natural event count, we use a pragmatic proxy: mileage in thousands (milage_k), converted to an integer “count-like” variable (milage_k_count).
This section mainly demonstrates the Poisson GLM workflow and interpretation.

6.1 Model specification

We model the expected mileage count (in thousands) using a Poisson GLM with a log link:

milage_k_count ~ age + accident_bin + brand + fuel_type + transmission + ext_col + int_col

In a Poisson GLM with log link:

• Response: milage_k_count (non-negative integer proxy)
• Predictors: age, accident history, and categorical vehicle characteristics
• Link: log( E[milage_k_count] ) = linear predictor

So coefficients are interpreted multiplicatively:
exp(beta) is the factor change in the expected count for a one-unit increase (or dummy switch), holding other variables fixed.

6.2 Estimation and overall performance

We fit the Poisson GLM by sourcing src/06_model_glm_poisson.R, which: 1) loads the feature data,
2) creates milage_k_count,
3) fits a Poisson GLM with log link, and
4) adds predictions/residuals back to the data.

source(here::here("src", "06_model_glm_poisson.R"))

# compact coefficient table incl. significance stars
poisson_tbl <- broom::tidy(glm_poisson) |>
  dplyr::mutate(
    signif = dplyr::case_when(
      p.value < 0.001 ~ "***",
      p.value < 0.01  ~ "**",
      p.value < 0.05  ~ "*",
      p.value < 0.1   ~ ".",
      TRUE            ~ ""
    )
  )

poisson_tbl |>
  dplyr::select(term, estimate, std.error, statistic, p.value, signif) |>
  dplyr::slice(1:12)

## # A tibble: 12 × 6
##    term           estimate std.error statistic   p.value signif
##    <chr>             <dbl>     <dbl>     <dbl>     <dbl> <chr> 
##  1 (Intercept)      3.79    0.0291      130.   0         ***   
##  2 age              0.0676  0.000408    166.   0         ***   
##  3 accident_bin     0.215   0.00453      47.5  0         ***   
##  4 brandAlfa       -0.564   0.0600       -9.39 5.92e- 21 ***   
##  5 brandAudi       -0.102   0.0191       -5.31 1.07e-  7 ***   
##  6 brandBentley    -0.941   0.0394      -23.9  6.01e-126 ***   
##  7 brandBMW        -0.161   0.0179       -8.99 2.48e- 19 ***   
##  8 brandBuick      -0.140   0.0303       -4.63 3.63e-  6 ***   
##  9 brandCadillac   -0.0474  0.0205       -2.31 2.08e-  2 *     
## 10 brandChevrolet  -0.233   0.0182      -12.9  7.88e- 38 ***   
## 11 brandChrysler   -0.209   0.0285       -7.32 2.45e- 13 ***   
## 12 brandDodge      -0.0818  0.0205       -3.99 6.72e-  5 ***

##        AIC pseudo_R2_deviance dispersion_pearson
## 1 79885.43          0.4922187            20.4108

If the dispersion is clearly above 1, the Poisson variance assumption (mean ≈ variance) may be violated (overdispersion). In a real use case you’d consider a quasi-Poisson or negative binomial model. We keep Poisson here for comparability and interpretation practice.

6.3 Interpretation of selected coefficients

In a Poisson GLM with log link:

• exp(beta) > 1 → expected count increases
  
• exp(beta) < 1 → expected count decreases
  
• percent change: (exp(beta) - 1) * 100

## # A tibble: 4 × 5
##   term               estimate rate_ratio pct_change   p.value
##   <chr>                 <dbl>      <dbl>      <dbl>     <dbl>
## 1 age                  0.0676      1.07         7   0        
## 2 accident_bin         0.215       1.24        24   0        
## 3 fuel_typeElectric   -1.01        0.365      -63.5 1.08e-190
## 4 transmissionManual  -0.318       0.728      -27.2 0

6.4 Residual diagnostics

6.5 Conclusions (Poisson GLM)

The Poisson GLM demonstrates how to model a count-like response with a log link and interpret effects as multiplicative changes via exp(beta). While milage_k_count is not a true event count, this workflow is useful to practice GLM estimation, interpretation, and basic diagnostics. In a real client setting with genuine count outcomes, overdispersion should be checked carefully and a quasi-Poisson or negative binomial model may be more appropriate.

7. Generalised Additive Model (GAM)

To relax the linearity assumption of the binomial GLM, we fit a Generalised Additive Model (GAM) to predict whether a car has had at least one reported accident (accident_bin). The GAM allows for non-linear relationships between key continuous predictors and accident probability, while retaining interpretable parametric effects for categorical variables.

7.1 Model specification

We model accident history using a GAM with a binomial family and logit link:

accident_bin ~ brand + fuel_type + transmission + cylinders + s(milage_k) + s(price_dollar) + s(horsepower) + s(age)

Where: - accident_bin is a binary response (0 = no reported accident, 1 = at least one reported accident). - brand, fuel_type, and transmission are categorical predictors. - cylinders enters the model linearly. - s(milage_k), s(price_dollar), s(horsepower), and s(age) are smooth spline functions capturing potential nonlinear effects. - The model is estimated using penalised likelihood with REML.

7.2 Model fitting and summary

Smoothness is selected automatically via REML, reducing the risk of overfitting despite flexible spline terms.

source(here::here("src", "07_model_gam.R"))

# Show a full summary of the model
summary(gam.car)

## 
## Family: binomial 
## Link function: logit 
## 
## Formula:
## accident_bin ~ brand + fuel_type + transmission + cylinders + 
##     s(milage_k) + s(price_dollar) + s(horsepower) + s(age)
## 
## Parametric coefficients:
##                           Estimate Std. Error z value Pr(>|z|)  
## (Intercept)             -8.419e-01  5.449e-01  -1.545   0.1223  
## brandAlfa               -1.006e+00  1.123e+00  -0.895   0.3705  
## brandAudi                1.611e-01  3.881e-01   0.415   0.6780  
## brandBentley            -3.536e-01  7.696e-01  -0.459   0.6459  
## brandBMW                -1.905e-01  3.710e-01  -0.513   0.6077  
## brandBuick               3.009e-01  6.049e-01   0.497   0.6189  
## brandCadillac            7.578e-03  4.240e-01   0.018   0.9857  
## brandChevrolet          -1.726e-01  3.858e-01  -0.447   0.6546  
## brandChrysler           -8.123e-02  5.774e-01  -0.141   0.8881  
## brandDodge               1.182e-01  4.339e-01   0.272   0.7853  
## brandFerrari             9.462e-01  1.290e+00   0.733   0.4634  
## brandFord               -6.184e-02  3.736e-01  -0.166   0.8685  
## brandGenesis            -1.085e+00  1.114e+00  -0.974   0.3303  
## brandGMC                 2.126e-01  4.378e-01   0.486   0.6272  
## brandHonda              -2.836e-01  4.795e-01  -0.591   0.5543  
## brandHummer              4.499e-01  6.215e-01   0.724   0.4692  
## brandHyundai             9.541e-02  4.502e-01   0.212   0.8322  
## brandINFINITI           -2.988e-01  4.638e-01  -0.644   0.5194  
## brandJaguar             -7.691e-02  5.063e-01  -0.152   0.8793  
## brandJeep               -3.293e-01  4.214e-01  -0.781   0.4346  
## brandKia                -1.080e-01  5.125e-01  -0.211   0.8331  
## brandLamborghini        -3.668e+01  1.861e+07   0.000   1.0000  
## brandLand               -7.975e-01  4.482e-01  -1.779   0.0752 .
## brandLexus               2.860e-01  3.964e-01   0.722   0.4706  
## brandLincoln            -1.300e-01  4.806e-01  -0.270   0.7868  
## brandMaserati           -1.126e+00  7.196e-01  -1.565   0.1177  
## brandMazda              -1.007e-01  5.153e-01  -0.195   0.8451  
## brandMercedes-Benz      -2.690e-02  3.773e-01  -0.071   0.9432  
## brandMINI               -1.305e+00  6.480e-01  -2.014   0.0440 *
## brandMitsubishi          4.321e-02  6.045e-01   0.071   0.9430  
## brandNissan              1.118e-01  4.120e-01   0.271   0.7861  
## brandPontiac            -2.782e-01  6.905e-01  -0.403   0.6870  
## brandPorsche             1.008e-01  4.148e-01   0.243   0.8080  
## brandRAM                 3.681e-01  4.656e-01   0.791   0.4292  
## brandRivian             -3.651e+01  1.733e+07   0.000   1.0000  
## brandSubaru              1.435e-01  4.535e-01   0.317   0.7516  
## brandTesla              -1.148e+00  9.016e-01  -1.273   0.2029  
## brandToyota             -1.912e-01  3.927e-01  -0.487   0.6263  
## brandVolkswagen         -4.975e-01  4.860e-01  -1.024   0.3060  
## brandVolvo              -5.360e-01  5.495e-01  -0.975   0.3294  
## fuel_typeE85 Flex Fuel  -1.764e-01  3.137e-01  -0.562   0.5740  
## fuel_typeElectric       -8.444e-01  6.694e-01  -1.261   0.2071  
## fuel_typeGasoline       -1.789e-01  2.573e-01  -0.696   0.4867  
## fuel_typeHybrid         -1.172e-01  3.416e-01  -0.343   0.7315  
## fuel_typePlug-In Hybrid  6.662e-01  5.035e-01   1.323   0.1858  
## transmissionManual      -1.408e-01  1.626e-01  -0.866   0.3866  
## cylinders                7.553e-03  5.476e-02   0.138   0.8903  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##                   edf Ref.df Chi.sq p-value    
## s(milage_k)     5.696  6.819 45.198 < 2e-16 ***
## s(price_dollar) 2.266  2.990 14.055 0.00284 ** 
## s(horsepower)   2.212  2.862  4.813 0.21621    
## s(age)          3.077  3.898  8.023 0.07770 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.102   Deviance explained = 11.1%
## -REML = 1623.1  Scale est. = 1         n = 3065

The summary output reports estimated degrees of freedom (EDF) and approximate significance tests for each smooth term.

7.3 Interpretation of smooth effects

The GAM summary reports estimated degrees of freedom (EDF) and significance tests for each smooth term, indicating whether nonlinear effects are present.

• Mileage (s(milage_k))
  Mileage shows a highly significant and strongly nonlinear effect on accident probability (EDF ≈ 5.7, p < 2e−16). The relatively large EDF indicates a complex relationship: accident risk does not increase at a constant rate with mileage. Instead, the effect varies across usage levels, reflecting changing exposure and driving patterns over a vehicle’s lifetime.

• Price (s(price_dollar))
  Price has a statistically significant smooth effect (EDF ≈ 2.3, p = 0.0028). While the effect is nonlinear, its lower EDF suggests a shape close to linear. Higher-priced vehicles tend to have a lower probability of reported accidents, consistent with newer vehicles, better maintenance, and improved safety features.

• Horsepower (s(horsepower))
  The smooth effect of horsepower is not statistically significant (p = 0.216). Although horsepower is modelled flexibly, the data provide no strong evidence that engine power independently affects accident probability once mileage, price, and other characteristics are controlled for.

• Age (s(age))
  The smooth effect of age is marginally insignificant (p = 0.078). This suggests that age alone does not strongly predict accident probability after accounting for mileage and vehicle characteristics. Much of the age-related risk appears to be captured indirectly through mileage.

Overall, mileage and price emerge as the most important nonlinear predictors of accident history in the GAM.

7.4 Interpretation of parametric effects

The parametric terms (brand, fuel type, transmission, and cylinders) are interpreted as in a standard logistic regression, holding all smooth effects constant.

• Brand effects
  Most brand coefficients are not statistically significant, indicating limited brand-specific differences in accident probability after controlling for mileage, price, and other variables. One exception is MINI, which shows a significantly lower accident probability compared to the reference brand (p = 0.044).

• Fuel type
  Fuel type coefficients are not statistically significant. Electric vehicles show a negative coefficient, suggesting lower accident odds, but this effect is not significant once nonlinear mileage and price effects are accounted for.

• Transmission
  Manual transmission does not have a statistically significant effect on accident probability in this model.

• Cylinders
  The number of cylinders enters linearly and is not statistically significant, suggesting little direct impact on accident risk.

Overall, categorical vehicle characteristics play a secondary role compared to usage-related variables such as mileage and price.

7.5 Conclusions (GAM)

The Generalised Additive Model identifies mileage as the dominant predictor of accident risk, showing a highly significant and nonlinear relationship with the probability of reported accidents. Vehicle price also matters: lower-priced cars are more likely to have had accidents. Other continuous predictors, including horsepower and age, show no strong independent effects once the nonlinear patterns of mileage and price are accounted for. Most categorical variables brand, fuel type, transmission, and cylinders are not statistically significant, suggesting that accident risk is driven primarily by usage intensity and vehicle value, rather than brand identity. Compared to the linear binomial GLM, the GAM captures nonlinear effects and provides a more nuanced and interpretable view of accident risk, although overall predictive power remains limited due to the inherent randomness of accidents.

SVM model results

We fit radial-kernel SVM regressors on log_price using both e1071::svm and kernlab (via caret::train). Both models use the same 80/20 train-test split; hyperparameters are tuned by cross-validation and evaluated on the hold-out test set below.

svm_metrics <- readr::read_csv(
  here("report", "models", "svm", "svm_log_price_metrics.csv"),
  show_col_types = FALSE
)

svm_best <- readr::read_lines(here("report", "models", "svm", "svm_best_model.txt"))[1]

svm_metrics_wide <- svm_metrics |>
  tidyr::pivot_wider(names_from = .metric, values_from = .estimate)

knitr::kable(svm_metrics_wide, digits = 3, caption = "Test metrics for SVM variants (target: log_price)")

Test metrics for SVM variants (target: log_price)

.estimator	model	rmse	mae	rsq
standard	e1071_radial	0.290	0.209	0.868
standard	kernlab_radial	0.354	0.251	0.803

The e1071 radial SVM is best by RMSE (~0.290) and R² (~0.868), outperforming the kernlab variant on this split. SVMs do not yield straightforward coefficient interpretations; they learn support vectors and decision functions in a transformed feature space. To understand feature effects you would rely on downstream tools (e.g., partial dependence or SHAP), but within this report we focus on comparative error metrics and note that the tuned radial kernel captures non-linear relationships beyond the linear/log models.

Cross-validation setup: both SVMs were tuned with 3-fold cross validation on the training set (same 80/20 split for both). e1071::tune() searched a compact grid of cost and gamma; caret::train(method = "svmRadial") searched a grid of C and sigma. Final metrics shown above are from the untouched test set, so cross validation was only for hyperparameter selection.

Best model: e1071_radial with RMSE = 0.2898 (lower is better)

Neural network results

Two regressors on log_price: a caret nnet (with dummying + scaling, 3-fold cross validation over size/decay) and a manual neuralnet (shallow hidden layer). Metrics below come from the test split; cross validation was used only for tuning.

nn_metrics <- readr::read_csv(
  here("report", "models", "nn", "nn_log_price_metrics.csv"),
  show_col_types = FALSE
)

nn_best <- readr::read_lines(here("report", "models", "nn", "nn_best_model.txt"))[1]

nn_metrics_wide <- nn_metrics |>
  tidyr::pivot_wider(names_from = .metric, values_from = .estimate)

knitr::kable(nn_metrics_wide, digits = 3, caption = "Test metrics for NN variants (target: log_price)")

Test metrics for NN variants (target: log_price)

.estimator	model	rmse	mae	rsq
standard	caret_nnet	0.369	0.265	0.791
standard	neuralnet_manual	0.408	0.285	0.743

Best NN model: caret_nnet with RMSE = 0.369 (lower is better) — the manual neuralnet edge is small (RMSE ≈ 0.388 vs. 0.392), so both nets are in a similar error band; the manual net slightly reduces bias on the high end (see fewer large positive residuals).